Resolvent and Scattering Matrix at the Maximum of the Potential
2000 Mathematics Subject Classification: 35P25, 81U20, 35S30, 47A10, 35B38. We study the microlocal structure of the resolvent of the semiclassical Schrödinger operator with short range potential at an energy which is a unique non-degenerate global maximum of the potential. We prove that it is a semiclassical Fourier integral operator quantizing the incoming and outgoing Lagrangian submanifolds associated to the fixed hyperbolic point. We then discuss two applications of this result...